\newproblem{lay:1_8_23}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.8.23}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by $f(x)=mx+b$.
	\begin{enumerate}[a.]
		\item Show that $f$ is a linear transformation when $b=0$.
		\item Find a property of linear transformations that is violated when $b\neq 0$.
		\item Why is $f$ called a linear function?
	\end{enumerate}
}{
  % Solution
	\begin{enumerate}[a.]
		\item We need to show that $\forall x_1,x_2\in\mathbb{R}$, $\forall c\in\mathbb{R}$
			\begin{itemize}
				\item $f(x_1+x_2)=f(x_1)+f(x_2)$ \\
				      In this particular case:
							\begin{center}
								$f(x_1+x_2)=m(x_1+x_2)=mx_1+mx_2=f(x_1)+f(x_2)$
							\end{center}
				\item $f(cx_1)=cf(x_1)$ \\
				      In this particular case:
							\begin{center}
								$f(cx_1)=m(cx_1)=c(mx_1)=cf(x_1)$
							\end{center}
			\end{itemize}
		\item When $b\neq 0$ none of the two properties is fulfilled. Let's see an example with the second one:\\
					\begin{center}
						$f(cx_1)=m(cx_1)+b=cmx_1+b\neq cmx_1+cb=c(mx_1+b)=cf(x_1)$
					\end{center}
	  \item $f$ is called a linear function because its graph $(x,f(x))$ is a line the 2D plane. However, to be a linear transformation the line needs to pass through
		      the origin. If $b\neq 0$ the line defined by $f$ does not pass through the origin.
	\end{enumerate}
}
\useproblem{lay:1_8_23}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
